The Old Man and the Sea and "A Clean, Well-Lighted Place" are alike in focusing their attention on the dignity of a poor, elderly man. In Old Man, this is Santiago, who fights the good and shows his manhood in bringing the giant marlin to shore even though the sharks manage to eat all its flesh. He shows what it is to live with courage and integrity and that it is how one lives, not what one gains, that matters.
Similarly, "A Clean, Well-Lighted Place" shows that having a clean, well lit cafe where he can sit and drink in peace is important to allowing an old man to live in a decent, dignified way. While the younger of two waiters wants to shut the cafe so he can go home, saying the old man could go to a bar, the older, wiser waiter knows it is important to allow the man his time in the quiet, clean cafe because what matter is how one lives, not simply gaining a drink in a dirty, crowded place.
The two works differ in that being a short story, "A Clean, Well-Lighted Place" does not offer the depth of character study that emerges in The Old Man in the Sea. Furthermore, "A Clean, Well-Lighted Place" shows the old man character from afar, with the older waiter as the point-of-view character. The older waiter is a focal point and is revealed as compassionate in his sensitivity to the needs of another. In contrast, Old Man is told from Santiago's point of view and primarily gives us access to his thoughts, not those of an observer.